We wrote the following essay on the Unity and Diversity of Logic, together with Roman Kossak, a few months ago.
It appeared as a mathematical introduction to the book we edited (also with Åsa Hirvonen and Juha Kontinen) Logic Without Borders (with essays by S. Abramsky, J. T. Baldwin, J. Burgess, X. Caicedo, Z. Chatzidakis, C. Di Prisco, M. Dzamonja, C. Franks, P. Galliani, L. Hella+J. Väänänen, W. Hodges, J. Hubicka+J. Nešetřil, T. Hyttinen, R. Kaye+T. Lok Wong, J. Kennedy, J. Kontinen, S. Lindell+H. Towsner+S. Weinstein, M. Malliaris+S. Shelah, I. Niiniluoto, J. Paris+A. Vencovská, S. Shelah and J. Väänänen.)
A couple of months just spent in Helsinki, with various conferences since May and many mathematical encounters, convinced me more than ever of the importance of the Unity of Logic viewpoint.
Here is the first paragraph of our essay:
What is mathematical logic today? How does it connect with its historical roots? How does it continue to serve as foundations of mathematics, and how does it impact mathematics in general? Does it continue to serve as the foundations of mathematics at all? What distinguishes advanced areas of mathematical logic from other branches of mathematics? What parts of mathematical logic should be considered philosophy, and what parts evolved into independent subdisciplines of algebra, analysis or computer science? The article by Juliette Kennedy in this volume addresses some of these issues directly, as does Jouko Väänänen’s personal account of the development of his interests in mathematical logic. Other articles in the volume might be construed as providing partial responses to these questions, of course not necessarily in a direct way, but through the connections and links they explore, both internally within logic and externally between logic and other disciplines.
You can download the essay from here.
Addenda: Javier Moreno has now read our essay. He seems to find it interesting (he suggested the topic is good for a book!) but found it too short, too dispersed and lacking a unified voice. (All of this I lift from a twitter conversation…)
To this I have to say:
- first of all, thanks Javier for reading!
- second, I agree it is too short (but as it was the introduction to a quite long volume, we didn’t want it to become like another article – it should somehow open up the question of unity versus diversity in logic today – but should not have the weight of the real papers collected – we are editors, not authors!)
- furthermore, I agree: it lacks unity! As it is the product of two minds, of two voices, of two points of view, it has a combination of both. Although we speak quite a lot with Roman (on logic, math, art and many other things), in the subject of our introduction there are points of disagreement (or different perspectives). At some point, the essay was going to be a conversation but it felt a bit overacted – we ended up doing write-and-rewrite of our own sentences, crisscrossing ideas. The result is bound to be pointing in at least two directions… I kind of like it that way at this point…
- there is a long essay, somehow on the same topic, and definitely recommended to anyone interested in the topic, by Jouko Väänänen, in the volume itself. We asked him to write his own statement, his own “manifesto” on why logic (and not a part of logic, or as is so fashionable, seeing logic as some part of geometry). The text he wrote is a superb piece of intellectual understanding of what logic is today, and may be.
- finally, I have been writing a longer piece for a volume for the Simplicity meeting – now almost finished. And Roman has written longer pieces on subjects connected to this (and we both have to write the mathematical parts of our joint project with artists Wanda Kossak and María Clara Cortés).